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16
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 219 (18 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
Solving Quadratic (0,1)Problems by Semidefinite Programs and Cutting Planes
, 1996
"... We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moder ..."
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Cited by 55 (7 self)
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We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner.
Nonpolyhedral Relaxations of GraphBisection Problems
, 1993
"... We study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing node and edgeincidence vectors of cuts. We prove that both relaxations provide the same bound. The main fact we prove is that the duality be ..."
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Cited by 37 (8 self)
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We study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing node and edgeincidence vectors of cuts. We prove that both relaxations provide the same bound. The main fact we prove is that the duality between the relaxed edge and nodevectors preserves very natural cardinality constraints on cuts. We present an analogous result also for the maxcut problem, and show a relation between the edge relaxation and some other optimality criteria studied before. Finally, we briefly mention possible applications for a practical computational approach.
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 32 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 25 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
Minimizing Breaks by Maximizing Cuts
, 2000
"... We propose to solve the break minimization problem in sports scheduling by transforming it into a maximum cut problem in an undirected graph and applying a branchandcut algorithm. Our approach outperforms previous approaches with constraint programming and integer programming techniques. ..."
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Cited by 20 (2 self)
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We propose to solve the break minimization problem in sports scheduling by transforming it into a maximum cut problem in an undirected graph and applying a branchandcut algorithm. Our approach outperforms previous approaches with constraint programming and integer programming techniques.
A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations
 In The sharpest cut, MPS/SIAM Ser. Optim
, 2001
"... The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 01 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with re ..."
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Cited by 20 (3 self)
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The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 01 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 01 programming problems by oddcycle inequalities. We also consider separating oddcycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for maxcut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.
Node and Edge Relaxations of the MaxCut Problem
, 1994
"... We study an upper bound on the maxcut problem defined via a relaxation of the discrete problem to a continuous nonlinear convex problem, which can be solved efficiently. We demonstrate how far the approach can be pushed using advanced techniques from numerical linear algebra and nonsmooth optimizat ..."
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Cited by 6 (2 self)
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We study an upper bound on the maxcut problem defined via a relaxation of the discrete problem to a continuous nonlinear convex problem, which can be solved efficiently. We demonstrate how far the approach can be pushed using advanced techniques from numerical linear algebra and nonsmooth optimization. Various classes of graphs with up to 50,000 nodes and up to four million edges are considered. Since the theoretical bound can be computed only with a certain precision in practice, we use duality between node and edgeoriented relaxations to estimate the difference between the theoretical and the computed bounds.